Recently I have wondered if it is possible to calculate the probability of winning a poker tournament based on which strategy you use and how your all in moves are distributed. In this article I will share with you my findings. Hope you have the patience to read it all the way through:-). The table below (the culmination of way too many hours of research) is a teaser of what is ahead if you continue reading.

Just to remind you, in my first post in this series I discovered an exponential relationship between the time an online freezeout tournament has been running and the percentage of players exited from the tournament. This enabled me to estimate both the time needed to reach the final table of a tournament given the number of players registered and what size tournament you should choose given the time you have available to play.

I needed this relationship to be able to estimate the number of players entering a tournament of a given duration and thus the total amount of chips in play. Once I know the total amount of chips in play is known, I can calculate the number of successive all in wins needed to win all the chips in the tournament. Finally, the last step in my calculations will be to set up different all in probability distributions to end up with the probability of winning an online poker tournament. Confused? Don’t worry:-) I will guide you through my calculations step by step in the remainder of this article. If you don’t like math, simply scroll down to the final table where I summarize my most important findings.

The table below summarizes the total amount of chips in play for different tournament durations and the successive all ins you will need to win to win the tournament:

Tournament duration [hours]

Number of Players

Total Chips in play (starting stack 1500)

Successive all ins needed to win (rounded numbers)

1

9

13500

3

2

60

90000

6

3

140

210000

7

4

340

510000

8

5

680

1020000

9

6

1110

1665000

10

Again, in order to keep things simple I assumed that each all in would double the 1500 chip starting stack. According to this assumption, 2 successive all in wins will increase your stack from 1500 to 6000, 3 successive all inn wins from 1500 to 12000 and so on.

To make things a bit more realistic, let’s assume that you win half the chips you need to win the tournament by making your opponent fold (i.e no show down). In this case the number of successive all ins needed to win is reduced by 1 for each of the tournament durations shown above.

I have chosen the following scenarios that IMO cover the typical all in situations you will experience during a tournament.

The simple reference calculation

Setting probability calculations aside and assuming all the players in the tournament (including yourself) have an equal chance of winning it, you will win a tournament with x registered players 1 out of x times. This means you will win a 100 player tournament 1 out of 100 times, a 200 player tournament 1 out of 200 times and so on. Obviously you should aim higher than this otherwise your bankroll will hit zero in no time.

1st scenario: On a roll

You have an 80% probability of winning all your hands.

If for example you need to win 3 successive all ins to win the tournament, the probability of winning it is 0,8*0,8*0,8 = 0,5 corresponding to 1 out of 2 tournaments.

2nd scenario: Coinflip

You have a 50% probability of winning all your hands.

3rd scenario: Underdog

You have a 30% probability of winning all your hands.

4th scenario: Realistic?

20% of your hands you are underdog (30% probability)

40% of your hands you have a coinflip (50% probability)

40% of your hands you are favorite (80% probability)

The table below summarizes the probability calculations for the 4 scenarios:

Duration[hours]

Reference

win rate

[1 out of ..]

On a roll

win rate

[1 out of..]

Coinflip

win rate

[1 out of ..]

Underdog

win rate

[1 out of..]

Realistic

win rate

[1 out of..]

1

9

2

8

37

na

2

60

3

32

412

16

3

140

4

64

1372

28

4

340

5

128

4572

50

5

680

6

256

15242

87

6

1110

7

512

50805

152

Interestingly, it turns out that in all but one scenario (the underdog) your tournament win rate is significantly larger than the simple reference calculation. In conclusion I think I have successfully managed to give a qualified estimate of the probabilities of winning online poker tournament. In my next article I will try to make some use of the numbers I have come up with. An obvious approach would be to calculate the expected return on investment (ROI) for each of the scenarios listed above.

I would greatly appreciate any comments on the math and final numbers.

I suppose this would all depend on whether your particular site is rigged for, or against you.

The significance of all the variables makes this pretty hard to really do. If you tell everyone that if they enter a tournament with 500 people, their chances of winning are 1/500, you will probably do just fine. It might not be true for everyone, but for the majority of people it is pretty close.

Thanks for the efforts you made, it appears to be useful as you can see how your style of playing will affect your win rate, although I still like to think that my chances of winning at 1/number of players.

like: this means,for example, that if in a 9 player freezout you play online all-in hands (that you consider, correctly, to be a 80% favourite), you should (in the long run) win 1 in every 2 tournaments. so, if 1st prize payout would be 75$ (for 15$ buy-in) you should get on average around 60$ net every 2 games, leading to a 30$/hour payout :p

what do you do for a living mark? would you be interested in co-authoring a book on poker? (you can write me in private).

Regards,
Lucian

(from Europe)

RaphJune 12, 2011

Hi Mark,

Great article, and research! I really enjoyed reading it. However, I have to agree with some of the other comments that the chances of winning a poker tournament are 1 out of X (X being the number of entrants). That is a far more realistic assumption because you must take into consideration that for a player to double up 10 times in a row (assuming they were dealt pocket aces and they held up every time) their opponents MUST have an equal size stack during every one of those double ups. Otherwise the player could be dealt pocket aces 200 times in a row, and he/she still would NOT have knocked out everyone in the tournament. This probably sounds confusing so allow me to attempt to illustrate it:

The website has rigged the play for you to be dealt pocket aces every hand while one opponent at YOUR (particular) table will always be dealt KK (so they call your all-in) and the board has been rigged so they can never win ;0)

Hand #1
——–
You go all in pre-flop with your AA, and are called by a player holding KK.

Total chips in play during this hand = 3,000 of your chips + 3,000 of your opponent’s chips = 6,000 chips

The board reads: Flop = A K 2, Turn = A, River = 3

Your aces held up, and now you have 6,000 chips; of course if your aces didn’t hold up you would have been knocked out of the tournament.

Hand #2
——-
Same all in by you, and you get called, and win again. Now you have your original 6,000 chips + 3,000 from your opponent = 9,000 chips

Total time elapsed = 6 minutes into the tournament thus far, and 990 players remaining (some knocked out by other players).

Therefore, even after having gone all in with aces, and winning every hand, the tournament would STILL last for hours, and your stack would only increase gradually because no one else would have as many chips as you, and you would only be able to win what they have.

In conclusion, you would have to repeat your all-in for SEVERAL HUNDREDS of hands (not merely 10). Some players would have been busted out by other players, and the tournament would still last a few hours. Therefore, the best answer to the probability of winning a (non-rigged) tournament is still 1 out of 1,000 players, or 0.1%.

Thanks for your comment! I agree that the underlying assumption of opponents you double up against having an equal stack to yours during all phases of the tournament, is not that realistic. If you noticed, I also assumed that half of the chips you accumulate come from making your opponents fold (i.e. no showdown). The model can of course be made better by tweaking the numbers above and including a parameter, which takes care of the fact that your opponents do not always have a same stack size as yours. Still I’m pretty sure the chances of winning a tournament are better than 1/xx. I your model for example, you do not take into account that the opponents you are up against also accumulate chips during the tournament. This greatly reduces the number of all ins in a row you have to win. I guess one way of improving the model would be to investigate how the average number of chips increases during a tournament.